where each zksubscriptzkz_{k} is a rational function of xisubscriptxix_{i} and yjsubscriptyjy_{j} (element
of F⁢(x1,…,xn,y1,…,yn)Fsubscriptx1normal-…subscriptxnsubscripty1normal-…subscriptynF(x_{1},\ldots,x_{n},y_{1},\ldots,y_{n})), is possible iff nnn is a
power of 222.

Remark. The form of Pfister’s theorem is stated in a way
so as to mirror the form of Hurwitz theorem. In fact, Pfister
proved the following: if FFF is a field and nnn is a power of 2,
then there exists a sum of squares identity of the form

Conversely, if nnn is not a power of 222, then
there exists a field FFF such that the above sum of square identity
does not hold for anyzi∈F⁢(x1,…,xn,y1,…,yn)subscriptziFsubscriptx1normal-…subscriptxnsubscripty1normal-…subscriptynz_{i}\in F(x_{1},\ldots,x_{n},y_{1},\ldots,y_{n}). Notice that zisubscriptziz_{i} is no longer
required to be a linear function of the yjsubscriptyjy_{j} anymore.

When FFF is the field of reals ℝℝ\mathbb{R}, we have the following
generalization, also due to Pfister:

Theorem.

If f∈ℝ⁢(X1,…,Xn)fℝsubscriptX1normal-…subscriptXnf\in\mathbb{R}(X_{1},\ldots,X_{n}) is positive semidefinite, then
fff can be written as a sum of 2nsuperscript2n2^{n} squares.

The above theorem is very closely related to Hilbert’s 17th Problem:

Hilbert’s 17th Problem.Whether it is possible, to
write a positive semidefinite rational function in nnnindeterminates over the reals, as a sum of squares of rational
functions in nnn indeterminates over the reals?

The answer is yes, and it was proved by Emil Artin in 1927.
Additionally, Artin showed that the answer is also yes if the reals
were replaced by the rationals.

I have seen that Hurwitz theorem is used most often, instead of Hurwitz's theorem. Perhaps it is because the z at the end of the name is similar sounding to s. On the other hand, I have never seen Pfister's theorem being stated as Pfister theorem.

It's impossible to pronounce the genitive marker "s" in the names ending in a sibilant phoneme as in Hurwitz, Weierstrass, Bush. Therefore the "s" letter must not be written in such names, but if one wants to denote the genitive of them then write e.g. Hurwitz' theorem. Otherwise the "s" letter may be written if it's an established, standard procedure.

I would like then to ask: does possesiveness affect the meaning of the notion? Originally I submitted a correction to the entry "Simmon's line" for forgetting 's in the phrase like "the line *** is a Simson line for triangle ***"

and I got a reply like 'a' makes it possible to write "Simson line" without 's. So, is it really true that the absence of 's make the notion more general meaning...

I am not sure, but it seems to me now that things like "Simson line", or "Green function", or "Prandtl number" are NOT some fixed 'line', 'function', 'number' and they just have the names of Simson, Green and Prandtl. And that's why one shouldn't use posseive case here.

On the other hand "Fermat's theorem", "Zorn's lemma", "Planck's constant" are some concrete "theorem", "lemma" and "constant" and they actually BELONG to the persons who invented them and thus possesive case should be used here.

What do you think about this?

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knowledge can become a science
only with a help of mathematics

ok I think I misunderstood your correction then, I'm sorry
I thought you were complaining about the "THE"
as in "THE line", because there are many of them, not a single one, that's why I changed it to "A" and closed the correction.

> ok I think I misunderstood your correction then
Wait, wait! I am not so sure that you misunderstood it ;).

For everyone to understand what we are talking about here is the original sentence which I thought has some problems:

"In the picture, the line passing through $U,V,W$ is the Simson line for $\triangle ABC$."

So, as already said "Simson line" is NOT some fixed line but rather a line determined by some things (see the entry). Thus "the Simson line for triangle..." sounds like triangle has ONE Simson line, which is not the case, so it is better to use "a Simson line".

But what we are talking about now, is another thing: when to use 's? According to what I wrote in the previous post "Simson line" should be written everywhere WITHOUT 's. The big question is of cource, whether what I wrote is actually correct ;). Have you really seen the expression "Simson's line"?
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knowledge can become a science
only with a help of mathematics

ok I think you had two points to mak in the correction, yet I failed to see one of them, I truly never understood that you were talking about posesives
f
G -----> H G
p \ /_ ----- ~ f(G)
\ / f ker f
G/ker f

But anyway, let's forget about that correction ;) and return to the issue of this post. So, once more: have you seen in some quite reliable source (I mean say, written by native speaker ;) "Simson's line" really WITH 's?

And what do you think about the following rule (retyped from previous posts):

-------------
It seems that things like "Simson line", or "Green function", or "Prandtl number" are NOT some fixed 'line', 'function', 'number' and they just have the names of Simson, Green and Prandtl. And that's why one shouldn't use posseive case here.

On the other hand "Fermat's theorem", "Zorn's lemma", "Planck's constant" are some concrete "theorem", "lemma" and "constant" and they actually BELONG to the persons who invented them and thus possesive case should be used here.
-------------

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knowledge can become a science
only with a help of mathematics

I did some research about this a long time ago, then I discovered I'm not supposed to write
Pythagoras's theorem but Pythagoras' theorem, but this was already pointed on this thread, otherwise I'm of not much help, but I got a few books behind me and this are some random checks. When nothing else is stated, they were taken from the subject index:

> it seems that when refering to a mathematical entity,
> [one] doens't uses 's (since mathematicians do not "own" math objects)
> but when refering to statements, lemmas and theorems,
> [one] does use 's

> That convention makes sense to me, so I would say
> "By the Simson's theorem on Simson lines..."

Well, it is something similar to what I said. It would be interesting to find some place where some rule is explicitly stated ;)

> I'll do more checks soon...

I'll also be looking for...
-------------------------------
knowledge can become a science
only with a help of mathematics